Question: Given $ m \angle CBD = 9x + 56$, $ m \angle ABC = 4x + 14$, and $ m \angle ABD = 83$, find $m\angle CBD$. $B$ $A$ $D$ $C$
Explanation: From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Substitute in the expressions that were given for each measure: $ {4x + 14} + {9x + 56} = {83}$ Combine like terms: $ 13x + 70 = 83$ Subtract $70$ from both sides: $ 13x = 13$ Divide both sides by $13$ to find $x$ $ x = 1$ Substitute $1$ for $x$ in the expression that was given for $m\angle CBD$ $ m\angle CBD = 9({1}) + 56$ Simplify: $ {m\angle CBD = 9 + 56}$ So ${m\angle CBD = 65}$.